Universal instantiation

Rule of inference in predicate logic
Universal instantiation
TypeRule of inference
FieldPredicate logic
Symbolic statement ∀ x A ⇒ A { x ↦ t } {\displaystyle \forall x\,A\Rightarrow A\{x\mapsto t\}}
Transformation rules
Propositional calculus
Rules of inference
  • Implication introduction / elimination (modus ponens)
  • Biconditional introduction / elimination
  • Conjunction introduction / elimination
  • Disjunction introduction / elimination
  • Disjunctive / hypothetical syllogism
  • Constructive / destructive dilemma
  • Absorption / modus tollens / modus ponendo tollens
  • Negation introduction
Rules of replacement
  • Associativity
  • Commutativity
  • Distributivity
  • Double negation
  • De Morgan's laws
  • Transposition
  • Material implication
  • Exportation
  • Tautology
Predicate logic
Rules of inference
  • Universal generalization / instantiation
  • Existential generalization / instantiation

In predicate logic, universal instantiation[1][2][3] (UI; also called universal specification or universal elimination,[citation needed] and sometimes confused with dictum de omni)[citation needed] is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.

Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."

Formally, the rule as an axiom schema is given as

∀ x A ⇒ A { x ↦ t } , {\displaystyle \forall x\,A\Rightarrow A\{x\mapsto t\},}

for every formula A and every term t, where A { x ↦ t } {\displaystyle A\{x\mapsto t\}} is the result of substituting t for each free occurrence of x in A. A { x ↦ t } {\displaystyle \,A\{x\mapsto t\}} is an instance of ∀ x A . {\displaystyle \forall x\,A.}

And as a rule of inference it is

from ⊢ ∀ x A {\displaystyle \vdash \forall xA} infer ⊢ A { x ↦ t } . {\displaystyle \vdash A\{x\mapsto t\}.}

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."[4]

Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.[5]

See also

References

  1. ^ Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375.[page needed]
  2. ^ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
  3. ^ Moore and Parker[full citation needed]
  4. ^ Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
  5. ^ Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press. OCLC 728954096. Here: p. 366.